Proving where a multivariable function is continuous

So I am fairly certain I know how to show the function is discontinuous at :

If was continuous at , then . However, if we approach along the parabola , then we have that since as gets sufficiently closer and closer to the origin. Therefore, is discontinuous at .

Is that proof correct, and if not, what is wrong?

Moving to the next portion, I am stuck; how do I prove that for all other points, is continuous. I have a general idea that I need to check points outside the region , inside that region, and then on the boundaries of the region (namely points along and . I am having trouble formalizing this into a proof though. Thanks for any help.

Edit: Er, stupid me, meant , not , otherwise what I wrote doesn't make sense.